with a compression check you just thread the hose with guage into the spark plug whole then crank then motor (cutting off fuel supply first) to see how much compression it makes, i believe all compression test guages have a peak hold feature so that you can see what the max compression(psi) the cylinder has...

a leakdown i think you use an air compressor and pump air into the cylinder and the guage measures how much pressure is bleading off (a leak). this is actually a better test then a compression because it gives you a better idea of what kinda condition the internals are in.